It’s possible to do via extensions now. Hyperweb is an extension that allows for adding atmos search engines.

I've been using hyperweb for a couple of months now. Absolutely outstanding app -- it has completely transformed my mobile web experience. Custom ad blocking filters, automatically opening reddit links in Narwhal, custom css so I can browse HN with a dark theme... I kind of look at it as a single app that turns Safari into a slightly shittier Firefox. And since Apple has neutered browsers on iOS, this is absolutely as good as it gets.

"Hyperweb..."

Wow nice! Now if there was a containers extension I could switch to Safari and away from Firefox.

Tab groups might do this since one of them is private browsing. If groups aren't isolated, implying they are by putting private browsing in there would be a huge UX fail.

I believe private browsing tabs in Safari have been isolated to some extent for a while already — if you sign into something in one tab, you aren't signed in in the other tab, suggesting separate cookies.